R Notebook

Here I experiment with combining flow fields with images (dithering).

Load packages:

library(ggnewscale)
library(imager)

Loading required package: magrittr

Attaching package: ‘imager’

The following object is masked from ‘package:magrittr’:

    add

The following objects are masked from ‘package:stats’:

    convolve, spectrum

The following object is masked from ‘package:graphics’:

    frame

The following object is masked from ‘package:base’:

    save.image

library(sf)

Linking to GEOS 3.9.1, GDAL 3.2.1, PROJ 7.2.1; sf_use_s2() is TRUE

library(tidyverse)

Registered S3 methods overwritten by 'dbplyr':
  method         from
  print.tbl_lazy     
  print.tbl_sql      
-- Attaching packages ------------------------------------------------------------------ tidyverse 1.3.1 --
v ggplot2 3.3.5     v purrr   0.3.4
v tibble  3.1.6     v dplyr   1.0.7
v tidyr   1.1.4     v stringr 1.4.0
v readr   2.1.1     v forcats 0.5.1
-- Conflicts --------------------------------------------------------------------- tidyverse_conflicts() --
x imager::add()       masks magrittr::add()
x stringr::boundary() masks imager::boundary()
x tidyr::extract()    masks magrittr::extract()
x tidyr::fill()       masks imager::fill()
x dplyr::filter()     masks stats::filter()
x dplyr::lag()        masks stats::lag()
x purrr::set_names()  masks magrittr::set_names()

Process image - snowscape

Read the image using imager::load.image():

#melville <- image_read("melville_lg.jpg")
snowscape <- load.image("snowscape.jpg")

Check the size of the image:

summary(snowscape)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.1686  0.4118  0.4195  0.5843  1.0000 

Convert image to data frame (still working with {imager}):

snowscape_df <- snowscape %>%
  as.data.frame(wide="c") %>% 
  mutate(color = rgb(c.1,
                     c.2,
                     c.3))

Convert to grayscale and to data frame:

snowscape_df_grayscale <- snowscape %>%
  grayscale() %>% 
  as.data.frame()

Join data frames with color and grayscale values:

snowscape_df <- snowscape_df %>%
  left_join(snowscape_df_grayscale,
            by = c("x", "y"))

Retrieve every nth row/column from image:

nx <- 10
ny <- 20

snowscape_sampled <- snowscape_df %>%
  filter(x %% nx == 1,
         y %% ny == 1) 

Plot:

snowscape_sampled %>%
  ggplot(aes(x, 
             y, 
             color = color)) + 
  geom_point(aes(size = value)) + 
  scale_color_identity() +
  scale_y_reverse() + 
  scale_size(range = c(1, 6)) +
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

ggsave("snowscape-circles.png")

Saving 7.29 x 4.51 in image

Plot with different shape:

snowscape_sampled %>%
  ggplot(aes(x, 
             y, 
             color = color)) + 
  geom_point(aes(size = value),
             shape = 17) +
  geom_point(size = 0.1,
             color = "black",
             shape = 20) +
  scale_color_identity() +
  scale_size(range = c(1, 5)) +
  scale_y_reverse() + 
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

ggsave("snowscape-dotted-diamonds.png")

Saving 7.29 x 4.51 in image

Retrieve every nth row/column from image:

nx <- 10
ny <- 20

snowscape_sampled <- snowscape_df %>%
  filter(x %% nx == 1,
         y %% ny == 1) 

Define length of segments:

l <- 15 # Experiment with the length

Create and plot the flow field:

n_min <- min(snowscape_sampled$x)
n_max <- max(snowscape_sampled$x)

# Offsets
x_o <- -max(snowscape_sampled$x)/2 # Experiment with the sign and the fraction
y_o <- 0 #max(snowscape_sampled$y)

df <- snowscape_sampled %>%
  mutate(angle = 1 * pi/6 * (1 -(x - x_o)/(n_max - n_min)),
         xend = (x + l * cos(angle)),
         yend = (y + l * sin(angle)))

df %>%
  ggplot() +
  geom_segment(aes(x = x,
                   y = y,
                   xend = xend,
                   yend = yend,
                   size = value,
                   color = color)) +
  scale_size(range = c(1, 8)) + # Experiment with the range
  scale_color_identity() +
  scale_y_reverse() +
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

ggsave("snowscape-flow-field.png")

Saving 7.29 x 4.51 in image

Process image - Meghan

Read the image using imager::load.image():

meghan <- load.image("meghan.jpg")

Check the size of the image:

summary(meghan)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.2980  0.6510  0.5623  0.8118  1.0000 

Convert image to data frame (still working with {imager}):

meghan_df <- meghan %>%
  as.data.frame(wide="c") %>% 
  mutate(color = rgb(c.1,
                     c.2,
                     c.3))

Convert to grayscale and to data frame:

meghan_df_grayscale <- meghan %>%
  grayscale() %>% 
  as.data.frame()

Join data frames with color and grayscale values:

meghan_df <- meghan_df %>%
  left_join(meghan_df_grayscale,
            by = c("x", "y"))

Crop image:

meghan_df <- meghan_df %>%
  filter(x <= 1250, y <= 1250)

Retrieve every nth row/column from image:

nx <- 20
ny <- 20

meghan_sampled <- meghan_df %>%
  filter(x %% nx == 1,
         y %% ny == 1) 

Plot:

meghan_sampled %>%
  ggplot() + 
  geom_point(aes(x, 
                 y,
                 color = value),
             shape = 15,
             size = 6) +
  scale_color_fermenter(palette = "greens") +
  new_scale_color() +
  geom_point(aes(x, 
                 y, 
                 color = color,
                 size = value)) + 
  scale_color_identity() +
  scale_y_reverse() + 
  scale_size(range = c(0.1, 3)) +
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

Warning in pal_name(palette, type) : Unknown palette greens

ggsave("meghan-circles.png",
       height = 8,
       width = 8,
       units = "in")

Plot with different shape:

meghan_sampled %>%
  ggplot() + 
  geom_point(aes(x,
                 y),
             color = "black",
             shape = 15,
             size = 6) +
  geom_point(aes(x, 
                 y,
                 color = color,
                 size = value),
             shape = 17) +
  # geom_point(aes(x, 
  #            y),
  #            size = 0.1,
  #            color = "black",
  #            shape = 20) +
  scale_color_identity() +
  scale_size(range = c(1, 4)) +
  scale_y_reverse() + 
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

ggsave("meghan-diamonds.png",
       height = 8,
       width = 8,
       units = "in")

Retrieve every nth row/column from image:

nx <- 25
ny <- 25

meghan_sampled <- meghan_df %>%
  filter(x %% nx == 1,
         y %% ny == 1) 

Define length of segments:

l <- 25 # Experiment with the length

Create and plot the flow field:

n_min <- min(meghan_sampled$x)
n_max <- max(meghan_sampled$x)

# Offsets
x_o <- 0 #max(meghan_sampled$x)/2 # Experiment with the sign and the fraction
y_o <- 0 #max(meghan_sampled$y)
# Initial rotation
init_r <- 1/8


df <- meghan_sampled %>%
  mutate(angle = ((x - x_o) * (y - y_o)) /(n_max - n_min)^2 * 2 * pi + init_r * pi,
         xend = (x + l * cos(angle)),
         yend = (y + l * sin(angle)))

df %>%
  ggplot() +
  geom_point(aes(x,
                 y, 
                 color = value),
             #color = "black",
             shape = 15,
             size = 6) +
  scale_color_fermenter(palette = "Greys") +
  new_scale_color() +
  geom_segment(aes(x = x,
                   y = y,
                   xend = xend,
                   yend = yend,
                   size = value,
                   color = color)) +
  scale_size(range = c(0.5, 3)) + # Experiment with the range
  scale_color_identity() +
  scale_y_reverse() +
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

ggsave("meghan-flow-field.png",
       height = 8,
       width = 8,
       units = "in")

Process image - Mariana

Read the image using imager::load.image():

mariana <- load.image("mariana-gato.jpg")

Check the size of the image:

summary(mariana)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.2784  0.4353  0.4709  0.6784  1.0000 

Convert image to data frame (still working with {imager}):

mariana_df <- mariana %>%
  as.data.frame(wide="c") %>% 
  mutate(color = rgb(c.1,
                     c.2,
                     c.3))

Convert to grayscale and to data frame:

mariana_df_grayscale <- mariana %>%
  grayscale() %>% 
  as.data.frame()

Join data frames with color and grayscale values:

mariana_df <- mariana_df %>%
  left_join(mariana_df_grayscale,
            by = c("x", "y"))

Crop image:

#mariana_df <- mariana_df %>%
#  filter(x <= 1250, y <= 1250)

Retrieve every nth row/column from image:

nx <- 30
ny <- 30

mariana_sampled <- mariana_df %>%
  filter(x %% nx == 1,
         y %% ny == 1) 

Plot:

mariana_sampled %>%
  ggplot() + 
  geom_point(aes(x, 
                 y,
                 color = value),
             shape = 15,
             size = 6) +
  scale_color_fermenter(palette = "Greys") +
  new_scale_color() +
  geom_point(aes(x, 
                 y, 
                 color = color,
                 size = value)) + 
  scale_color_identity() +
  scale_y_reverse() + 
  scale_size(range = c(0.1, 3)) +
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

# ggsave("meghan-circles.png",
#        height = 8,
#        width = 8,
#        units = "in")

Plot with different shape:

mariana_sampled %>%
  ggplot() + 
  geom_point(aes(x,
                 y),
             color = "black",
             shape = 15,
             size = 6) +
  geom_point(aes(x, 
                 y,
                 color = color,
                 size = value),
             shape = 17) +
  # geom_point(aes(x, 
  #            y),
  #            size = 0.1,
  #            color = "black",
  #            shape = 20) +
  scale_color_identity() +
  scale_size(range = c(1, 4)) +
  scale_y_reverse() + 
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

# ggsave("meghan-diamonds.png",
#        height = 8,
#        width = 8,
#        units = "in")

Retrieve every nth row/column from image:

nx <- 25
ny <- 25

mariana_sampled <- mariana_df %>%
  filter(x %% nx == 1,
         y %% ny == 1) 

Define length of segments:

l <- 25 # Experiment with the length

Create and plot the flow field:

n_min <- min(meghan_sampled$x)
n_max <- max(meghan_sampled$x)

# Offsets
x_o <- 0 #max(meghan_sampled$x)/2 # Experiment with the sign and the fraction
y_o <- 0 #max(meghan_sampled$y)
# Initial rotation
init_r <- 1/8


df <- mariana_sampled %>%
  mutate(angle = ((x - x_o) * (y - y_o)) /(n_max - n_min)^2 * 2 * pi + init_r * pi,
         xend = (x + l * cos(angle)),
         yend = (y + l * sin(angle)))

df %>%
  ggplot() +
  geom_point(aes(x,
                 y, 
                 color = value),
             color = "black",
             shape = 15,
             size = 6) +
  scale_color_distiller(palette = "Greys") +
  new_scale_color() +
  geom_segment(aes(x = x,
                   y = y,
                   xend = xend,
                   yend = yend,
                   size = value,
                   color = color)) +
  scale_size(range = c(0.5, 3)) + # Experiment with the range
  scale_color_identity() +
  scale_y_reverse() +
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

ggsave("mariana-flow-field.png",
       height = 8,
       width = 8,
       units = "in")

Process image - Oregon

Read the image using imager::load.image():

oregon <- load.image("oregon.jpg")

Check the size of the image:

summary(oregon)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.3137  0.4627  0.4681  0.6157  1.0000 

Convert image to data frame (still working with {imager}):

oregon_df <- oregon %>%
  as.data.frame(wide="c") %>% 
  mutate(color = rgb(c.1,
                     c.2,
                     c.3))

Convert to grayscale and to data frame:

oregon_df_grayscale <- oregon %>%
  grayscale() %>% 
  as.data.frame()

Join data frames with color and grayscale values:

oregon_df <- oregon_df %>%
  left_join(oregon_df_grayscale,
            by = c("x", "y"))

Retrieve every nth row/column from image:

nx <- 10
ny <- 10

oregon_sampled <- oregon_df %>%
  filter(x %% nx == 1,
         y %% ny == 1) 

Create and plot the flow field:

# Offsets
x_o <- 560
y_o <- 140


df <- oregon_sampled %>%
  #length of segments depends on position, and the angle is constant from x_o, y_o
  mutate(xend = (x + (x - x_o)/30 * pi/4),
         yend = (y + (y - y_o)/30 * pi/4))

df %>%
  ggplot() +
  geom_point(aes(x,
                 y,
                 color = value),
             shape = 15,
             size = 6) +
  scale_color_distiller(palette = "Greys") +
  new_scale_color() +
  geom_segment(aes(x = x,
                   y = y,
                   xend = xend,
                   yend = yend,
                   size = value,
                   color = color)) +
  scale_size(range = c(0.1, 2)) + # Experiment with the range
  scale_color_identity() +
  scale_y_reverse() +
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

ggsave("oregon-flow-field.png",
       height = 8,
       width = 8,
       units = "in")

Waldport

See https://twitter.com/mark_lawler/status/1490467641676337154/photo/2 for this image:

Read the image using imager::load.image():

waldport <- load.image("waldport.jpg")

Check the size of the image:

summary(waldport)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 0.0000  0.2235  0.4235  0.4616  0.6902  1.0000 

Convert image to data frame (still working with {imager}):

waldport_df <- waldport %>%
  as.data.frame(wide="c") %>% 
  mutate(color = rgb(c.1,
                     c.2,
                     c.3))

Convert to grayscale and to data frame:

waldport_df_grayscale <- waldport %>%
  grayscale() %>% 
  as.data.frame()

Join data frames with color and grayscale values:

waldport_df <- waldport_df %>%
  left_join(waldport_df_grayscale,
            by = c("x", "y"))

Retrieve every nth row/column from image:

nx <- 20
ny <- 20

waldport_sampled <- waldport_df %>%
  filter(x %% nx == 1,
         y %% ny == 1) 

Create and plot the flow field:

# Offsets
x_o <- -1000
y_o <- 770


df <- waldport_sampled %>%
  #length of segments depends on position, and the angle is constant from x_o, y_o
  mutate(xend = (x + (x - x_o)/20 * pi/8),
         yend = (y + (y - y_o)/20 * pi/8))

df %>%
  ggplot() +
  geom_point(aes(x,
                 y,
                 color = rev(y)),
             shape = 15,
             size = 6) +
  scale_color_gradientn(colors = MexBrewer::mex.brewer("Atentado")) +
  new_scale_color() +
  geom_segment(aes(x = x,
                   y = y,
                   xend = xend,
                   yend = yend,
                   size = value,
                   color = color)) +
  scale_size(range = c(0.01, 0.75)) + # Experiment with the range
  scale_color_identity() +
  scale_y_reverse() +
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

ggsave("waldport-flow-field.png",
       height = 8,
       width = 8,
       units = "in")

One more of another beach! Check this: https://twitter.com/GiuIianoDMedici/status/1490715140051947521

Read the image using imager::load.image():

beach <- load.image("beach.jpg")

Check the size of the image:

beach

Image. Width: 899 pix Height: 1199 pix Depth: 1 Colour channels: 3 

Convert image to data frame (still working with {imager}):

beach_df <- beach %>%
  as.data.frame(wide="c") %>% 
  mutate(color = rgb(c.1,
                     c.2,
                     c.3))

Convert to grayscale and to data frame:

beach_df_grayscale <- beach %>%
  grayscale() %>% 
  as.data.frame()

Join data frames with color and grayscale values:

beach_df <- beach_df %>%
  left_join(beach_df_grayscale,
            by = c("x", "y"))

Retrieve every nth row/column from image:

nx <- 10
ny <- 10

beach_sampled <- beach_df %>%
  filter(x %% nx == 1,
         y %% ny == 1) 

Create and plot the flow field:

# Offsets
x_o <- 200
y_o <- 480


df <- beach_sampled %>%
  #length of segments depends on position, and the angle is constant from x_o, y_o
  mutate(xend = (x + (x - x_o)/20 * pi/4),
         yend = (y + (y - y_o)/20 * pi/4))

df %>%
  ggplot() +
  geom_point(aes(x,
                 y,
                 color = rev(y)),
             shape = 15,
             size = 6) +
  scale_color_gradientn(colors = MexBrewer::mex.brewer("Atentado")) +
  new_scale_color() +
  geom_segment(aes(x = x,
                   y = y,
                   xend = xend,
                   yend = yend,
                   size = sqrt((x - x_o)^2 + (y - y_o)^2),
                   color = color)) +
  scale_size(range = c(0.5, 1.5)) + # Experiment with the range
  scale_color_identity() +
  scale_y_reverse() +
  coord_equal(expand = FALSE) +
  theme_void() +
  theme(legend.position = "none")

ggsave("beach-flow-field.png",
       height = 8,
       width = 8,
       units = "in")

Flower 1 and spirals

Read the image using imager::load.image():

flower <- load.image("flower.jpg")

Check the size of the image:

flower

Image. Width: 720 pix Height: 960 pix Depth: 1 Colour channels: 3 

Convert image to data frame (still working with {imager}):

flower_df <- flower %>%
  as.data.frame(wide="c") %>% 
  mutate(color = rgb(c.1,
                     c.2,
                     c.3))

Convert to grayscale and to data frame:

flower_df_grayscale <- flower %>%
  grayscale() %>% 
  as.data.frame()

Join data frames with color and grayscale values:

flower_df <- flower_df %>%
  left_join(flower_df_grayscale,
            by = c("x", "y"))

Convert the image to sf to use sf functions to find nearest features to borrow colors:

flower_sf <- flower_df %>%
    st_as_sf(coords = c("x", "y"))

Golden spiral:

#points <- 500

# Defining the Golden Angle
angle <- pi * (3 - sqrt(5))

#t_o <- seq(1, 10, 1) 

gs <- data.frame(t_o = seq(1, 750, 0.1) * angle) %>%
  mutate(x_o = 300 + t_o * sin(t_o),
         y_o = 300 + t_o * cos(t_o),
         x_end = x_o + 20 * sin(t_o),
         y_end = y_o + 20 * cos(t_o))

Convert to sf to borrow the colors from the image:

gs <- gs %>%
  mutate(x = x_o,
         y = y_o) %>%
  st_as_sf(coords = c("x", "y"))

Find the nearest feature in the original image to borrow the colors:

flower_colors <- flower_df[gs %>%
                                   st_nearest_feature(flower_sf),]

Bind the colors to the packed circles:

gs <- gs %>%
  mutate(color = flower_colors$color,
         value = flower_colors$value)

Plot:

# Make a scatter plot of points in a spiral
ggplot() +
  geom_point(data= flower_df,
             aes(x,
                 y,
                 color = value),
             #color = "black",
             shape = 15,
             size = 6) +
  scale_color_gradientn(colors = MexBrewer::mex.brewer("Atentado")) +
  #scale_color_distiller(palette = "Blues") +
  new_scale_color() +
  geom_segment(data = gs %>%
                 filter(x_o > 10 & x_o < 710,
                        y_o > 10 & y_o < 950),
               aes(x = x_o,
                   xend = x_end,
                   y = y_o,
                   yend = y_end,
                   size = t_o,
                   color = color)) +
  scale_y_reverse() +
  scale_color_identity() +
  scale_size(range = c(0.1, 3)) +
  coord_equal() +
  theme_void() +
  theme(legend.position = "none")

ggsave("flower-flow-field.png",
       height = 8,
       width = 8,
       units = "in")

Flower 2 and spirals

Read the image using imager::load.image():

flower <- load.image("flower-2.jpg")

Check the size of the image:

flower

Image. Width: 720 pix Height: 960 pix Depth: 1 Colour channels: 3 

Convert image to data frame (still working with {imager}):

flower_df <- flower %>%
  as.data.frame(wide="c") %>% 
  mutate(color = rgb(c.1,
                     c.2,
                     c.3))

Convert to grayscale and to data frame:

flower_df_grayscale <- flower %>%
  grayscale() %>% 
  as.data.frame()

Join data frames with color and grayscale values:

flower_df <- flower_df %>%
  left_join(flower_df_grayscale,
            by = c("x", "y"))

Convert the image to sf to use sf functions to find nearest features to borrow colors:

flower_sf <- flower_df %>%
    st_as_sf(coords = c("x", "y"))

Golden spiral:

#points <- 500

# Defining the Golden Angle
angle <- pi * (3 - sqrt(5))

#t_o <- seq(1, 10, 1) 

gs <- data.frame(t_o = seq(1, 750, 0.1) * angle) %>%
  mutate(x_o = 160 + t_o * sin(t_o),
         y_o = 310 + t_o * cos(t_o),
         x_end = x_o + 20 * sin(t_o),
         y_end = y_o + 20 * cos(t_o))

Convert to sf to borrow the colors from the image:

gs <- gs %>%
  mutate(x = x_o,
         y = y_o) %>%
  st_as_sf(coords = c("x", "y"))

Find the nearest feature in the original image to borrow the colors:

flower_colors <- flower_df[gs %>%
                                   st_nearest_feature(flower_sf),]

Bind the colors to the spiral:

gs <- gs %>%
  mutate(color = flower_colors$color,
         value = flower_colors$value)

Plot:

# Make a scatter plot of points in a spiral
ggplot() +
  # geom_point(data= flower_df,
  #            aes(x,
  #                y,
  #                color = value),
  #            #color = "black",
  #            shape = 15,
  #            size = 6) +
  # scale_color_gradientn(colors = MexBrewer::mex.brewer("Atentado")) +
  # scale_color_distiller(palette = "Blues") +
  # new_scale_color() +
  geom_segment(data = gs %>%
                 filter(x_o > 10 & x_o < 710,
                        y_o > 10 & y_o < 950),
               aes(x = x_o,
                   xend = x_end,
                   y = y_o,
                   yend = y_end,
                   size = t_o,
                   color = color)) +
  scale_y_reverse() +
  scale_color_identity() +
  scale_size(range = c(0.1, 3)) +
  coord_equal() +
  theme_void() +
  theme(legend.position = "none")

ggsave("flower-2-flow-field.png",
       height = 8,
       width = 8,
       units = "in")

Knots (nodal curves): see https://mathcurve.com/courbes2d.gb/noeud/noeud.shtml

Polar equation: \[ \rho = a \tan(n\theta) \] or: \[ \rho = a \cot(n\theta) \] with \(n>0\). Each curve is obtained for: \[ -\frac{\pi}{2n} < \theta < \frac{\pi}{2n} \]

Create a data frame to calculate the values of the knot function with parameters \(a\) and \(n\). If \(n\) is a rational number \(p/q\), the number of branches is \(2p\) if \(q\) is odd, and \(p\) if \(q\) is even. Each rotation of a branch is by an angle of \(\pi/n\):

# Scale factor
a <- 1
#Extent of values: larger ext means that fewer angles theta will be used in the calculation. If ext == 1 then the whole domain -pi/(2n) to pi/(2n) is used
ext <- 10 # Must be greater than 1
p <- 6
q <- 5
n <- p/q
# Number of branches
nb <- ifelse(q %% 2 == 1, 2 * p, p)

# Initialize data frame
df <- data.frame(t = seq(-pi/(ext * 2 * n), 
                         pi/(ext * 2 * n), 
                         length.out = 180)) %>%
  slice(3:178)

df <- df %>%
  # Rotation of branches is between 0 and 2 * pi (which replicates zero); the step increments are the 2 * pi divided by number of branches
  cbind(k = rep(seq(0, 2 * pi * (1 - 1/nb), by = 2 * pi/nb), each = nrow(df))) %>%
  # Calculate polar coordinates and then convert to cartesian coordinates
  mutate(r = a * tan(n * (t + k)),
         x = r * cos(t + k),
         y = r * sin(t + k)) #%>%
  # # Rotation
  # mutate(x = x * cos(k) - y * sin(k),
  #        y = x * sin(k) + y * cos(k))
  

Plot:

ggplot() +
  geom_point(data = df,
             aes(x,
                 y,
                 color = factor(k))) + 
  coord_equal()

Dopler spiral: see https://mathcurve.com/courbes2d.gb/doppler/doppler.htm

Parametric equations: \[ x = a(t\cos(t) + kt) \] and: \[ y = at\sin(t) \]

Create a data frame to calculate the values of the spiral with parameters \(a\) and \(k\):

# Scale factor
a <- 3
k <- 0.5

# Initialize data frame
df <- data.frame(t = seq(0, 
                         150, 
                         length.out = 300)) %>%
  # Calculate polar coordinates and then convert to cartesian coordinates
  mutate(x = a * (t * cos(t) + k * t),
         y = a * t * sin(t),
         x_end = x - 20 * cos(t),
         y_end = y + 20 * sin(t))

Plot:

ggplot() +
  geom_point(data = df,
             aes(x,
                 y,
                 color = t)) + 
  geom_segment(data = df,
             aes(x,
                 y,
                 xend = x_end,
                 yend = y_end)) +
  coord_equal()

Flowers and spirals

Read the image using imager::load.image():

flower <- load.image("flowers-titi.jpg")

Check the size of the image:

flower

Image. Width: 960 pix Height: 640 pix Depth: 1 Colour channels: 3 

Convert image to data frame (still working with {imager}):

flower_df <- flower %>%
  as.data.frame(wide="c") %>% 
  mutate(color = rgb(c.1,
                     c.2,
                     c.3))

Convert to grayscale and to data frame:

flower_df_grayscale <- flower %>%
  grayscale() %>% 
  as.data.frame()

Join data frames with color and grayscale values:

flower_df <- flower_df %>%
  left_join(flower_df_grayscale,
            by = c("x", "y"))

Convert the image to sf to use sf functions to find nearest features to borrow colors:

flower_sf <- flower_df %>%
    st_as_sf(coords = c("x", "y"))

Create a data frame to calculate the values of the spiral with parameters \(a\) and \(k\):

# Scale factor
a <- 0.25
k <- 0.1

# Initialize data frame
ds <- data.frame(t = seq(0, 
                         3500, 
                         length.out = 8000)) %>%
  # Calculate polar coordinates and then convert to cartesian coordinates
  mutate(x_o = 220 + a * (t * cos(t) + k * t),
         y_o = 460 + a * t * sin(t),
         x_end = x_o - t/200 * cos(t),
         y_end = y_o + t/200 * sin(t))

Convert to sf to borrow the colors from the image:

ds <- ds %>%
  mutate(x = x_o,
         y = y_o) %>%
  st_as_sf(coords = c("x", "y"))

Find the nearest feature in the original image to borrow the colors:

flower_colors <- flower_df[ds %>%
                                   st_nearest_feature(flower_sf),]

Bind the colors to the spiral:

ds <- ds %>%
  mutate(color = flower_colors$color,
         value = flower_colors$value)

Plot:

# Make a scatter plot of points in a spiral
ggplot() +
  geom_point(data= flower_df,
             aes(x,
                 y,
                 color = value),
             #color = "black",
             shape = 15,
             size = 6) +
  scale_color_gradientn(colors = MexBrewer::mex.brewer("Atentado")) +
  #scale_color_distiller(palette = "Blues") +
  new_scale_color() +
  geom_segment(data = ds %>%
                 filter(x_o > -10 & x_o < 970,
                        y_o > -10 & y_o < 650),
               aes(x = x_o,
                   xend = x_end,
                   y = y_o,
                   yend = y_end,
                   size = t,
                   color = color)) +
  scale_y_reverse() +
  scale_color_identity() +
  scale_size(range = c(1, 4)) +
  coord_equal() +
  theme_void() +
  theme(legend.position = "none")

ggsave("flower-titi-flow-field-atentado.png",
       height = 8,
       width = 8,
       units = "in")

Try this (see: https://www.facebook.com/mhernandezarrese/posts/10158652605032825):

And this by Mike Kukucska: